Simple string permutations questionThe answer is correct, but if you're writing it up for homework / an exam etc., you should add that $\binom{5}{2}$ is the number of ways to allocate the positions of the vowels in the sequence.

Fast way to get a position of combination (without repetitions)This question has been asked before multiple times on both this website and StackOverflow; one of the existing answers may suit you. (That's assuming that your $\left(\!\!\binom{n}{k}\!\!\right)$ is just an odd way of writing $\binom{n}{k}$ because I don't know what else it could mean.)

search on a split data structureCan you edit the question to make it clearer in the formatting which parts are quotation from the test / homework / question, and which parts are your own thoughts / questions?

Maximizing profit (dynamic programming)Guess you need to first read about dynamic programming before solving exercises. Did you manage to solve all (or most) of questions 1 to 18, before attempting question 19?

“What if” math joke: the derivative of $\ln(x)^e$@bcrist: Ah good point; perhaps one should say that though the What-If series contains references to the XKCD comics, there is none in the other direction. (Sort of how I can make references to a TV show, but a TV show will never make references to me.) At any rate, the subject of this question definitely doesn't count as "an xkcd", whatever that means. Nor is it even a joke, IMO.